Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $p = \dfrac{t^2 - 12t + 32}{-t - 8} \times \dfrac{t + 8}{-t + 4} $
Explanation: First factor the quadratic. $p = \dfrac{(t - 4)(t - 8)}{-t - 8} \times \dfrac{t + 8}{-t + 4} $ Then factor out any other terms. $p = \dfrac{(t - 4)(t - 8)}{-(t + 8)} \times \dfrac{t + 8}{-(t - 4)} $ Then multiply the two numerators and multiply the two denominators. $p = \dfrac{ (t - 4)(t - 8) \times (t + 8) } { -(t + 8) \times -(t - 4) } $ $p = \dfrac{ (t - 4)(t - 8)(t + 8)}{ (t + 8)(t - 4)} $ Notice that $(t + 8)$ and $(t - 4)$ appear in both the numerator and denominator so we can cancel them. $p = \dfrac{ \cancel{(t - 4)}(t - 8)(t + 8)}{ (t + 8)\cancel{(t - 4)}} $ We are dividing by $t - 4$ , so $t - 4 \neq 0$ Therefore, $t \neq 4$ $p = \dfrac{ \cancel{(t - 4)}(t - 8)\cancel{(t + 8)}}{ \cancel{(t + 8)}\cancel{(t - 4)}} $ We are dividing by $t + 8$ , so $t + 8 \neq 0$ Therefore, $t \neq -8$ $p = t - 8 ; \space t \neq 4 ; \space t \neq -8 $